![TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:1
PH?N T?CH H?I QUI TUY?N T?NH ??N GI?N
17.1 Ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh
Ph?n t¨ªch h?i qui tuy?n t¨ªch ??n gi?n (Simple Linear Regression Analysis) l¨¤
t¨¬m s? li¨ºn h? gi?a 2 bi?n s? li¨ºn t?c: bi?n ??c l?p (bi?n d? ?o¨¢n) tr¨ºn tr?c ho¨¤nh x
v?i bi?n ph? thu?c (bi?n k?t c?c) tr¨ºn tr?c tung y. Sau ?¨® v? m?t ???ng th?ng h?i
qui v¨¤ t? ph??ng tr¨¬nh ???ng th?ng n¨¤y ta c¨® th? d? ?o¨¢n ???c bi?n y (v¨ª d?: c?n
n?ng) khi ?? c¨® x (v¨ª d?: tu?i)
V¨ª d? 1: Ta c¨® 1 m?u g?m 6 tr? t? 1-6 tu?i, c¨® c?n n?ng nh? b?ng sau:
Tu?i C?n n?ng (kg)
1 10
2 12
3 14
4 16
5 18
6 20
N?i c¨¢c c?p (x,y) n¨¤y ta th?y c¨® d?ng 1 ph??ng tr¨¬nh b?c nh?t: y=2x+8
(trong ?¨® 2 l¨¤ ?? d?c v¨¤ 8 l¨¤ ?i?m c?t tr¨ºn tr?c tung y khi x=0). Trong th?ng k¨º
ph??ng tr¨¬nh ???ng th?ng (b?c nh?t) n¨¤y ???c vi?t d??i d?ng:
y= ?x + ? [1]](https://image.slidesharecdn.com/phan-tich-hoi-quy-tuyen-tinh-don-gian-150507163243-lva1-app6891/85/Phan-tich-hoi-quy-tuyen-tinh-don-gian-1-320.jpg)
![TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:3
kh?ng c¨°n l¨¤ 100% m¨¤ ch? c¨°n 97% v¨¬ c¨® s? sai l?ch t?i y1 v¨¤ y2. S? sai l?ch n¨¤y
trong th?ng k¨º g?i l¨¤ ph?n d? (residual) ho?c errors.
G?i y1, y2, y3, y4, y5, y6 l¨¤ tr? s? quan s¨¢t v¨¤ y¡¯1, y¡¯2, y¡¯3, y¡¯4, y¡¯5, y¡¯6 l¨¤ tr? s? ??c ?o¨¢n
n?m tr¨ºn ???ng h?i qui, ?1, ?2, ?3, ?4, ?5, ?6 l¨¤ ph?n d?.
Nh? v?y ?1= y1 ¨Cy¡¯1
?2 = y2 ¨Cy¡¯2
?3 = y3 ¨Cy¡¯3
?4 = y4 ¨Cy¡¯4
?5 = y5 ¨C y¡¯5
?6 = y6 ¨Cy¡¯6
Khi ?¨® ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh ???c vi?t d??i d?ng t?ng qu¨¢t nh? sau:
y¡¯= ¦Âxi + ?i+ ?i [2]
Nh? v?y n?u ph?n d? ?i c¨¤ng nh? s? li¨ºn h? gi?a x,y c¨¤ng l?n v¨¤ ng??c l?i. Ph?n
li¨ºn h? c¨°n ???i g?i l¨¤ ph?n h?i qui. M? h¨¬nh h?i qui tuy?n t¨ªch ???c m? t? nh? sau:
D? li?u= H?i qui (Regression) + Ph?n d? (Residual)
17.1.2 ??c t¨ªnh h? s? t??ng quan ? v¨¤ ch?n ?
Mu?n v? ???c ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh c?n ph?i ??c t¨ªnh ???c ?? d?c
? v¨¤ ch?n ? tr¨ºn tr?c tung.
V¨ª d? 3: N?u ch¨²ng ta ch?n m?t m?u th?c t? g?m 30 em t? 1-6 tu?i v¨¤ k?t qu? c?n
n?ng t??ng ?ng c?a 30 em ???c v? trong bi?u ?? sau:](https://image.slidesharecdn.com/phan-tich-hoi-quy-tuyen-tinh-don-gian-150507163243-lva1-app6891/85/Phan-tich-hoi-quy-tuyen-tinh-don-gian-3-320.jpg)
More Related Content
What's hot (20)
Similar to Phan tich-hoi-quy-tuyen-tinh-don-gian (20)
Phan tich-hoi-quy-tuyen-tinh-don-gian
- 1. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:1 PH?N T?CH H?I QUI TUY?N T?NH ??N GI?N 17.1 Ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh Ph?n t¨ªch h?i qui tuy?n t¨ªch ??n gi?n (Simple Linear Regression Analysis) l¨¤ t¨¬m s? li¨ºn h? gi?a 2 bi?n s? li¨ºn t?c: bi?n ??c l?p (bi?n d? ?o¨¢n) tr¨ºn tr?c ho¨¤nh x v?i bi?n ph? thu?c (bi?n k?t c?c) tr¨ºn tr?c tung y. Sau ?¨® v? m?t ???ng th?ng h?i qui v¨¤ t? ph??ng tr¨¬nh ???ng th?ng n¨¤y ta c¨® th? d? ?o¨¢n ???c bi?n y (v¨ª d?: c?n n?ng) khi ?? c¨® x (v¨ª d?: tu?i) V¨ª d? 1: Ta c¨® 1 m?u g?m 6 tr? t? 1-6 tu?i, c¨® c?n n?ng nh? b?ng sau: Tu?i C?n n?ng (kg) 1 10 2 12 3 14 4 16 5 18 6 20 N?i c¨¢c c?p (x,y) n¨¤y ta th?y c¨® d?ng 1 ph??ng tr¨¬nh b?c nh?t: y=2x+8 (trong ?¨® 2 l¨¤ ?? d?c v¨¤ 8 l¨¤ ?i?m c?t tr¨ºn tr?c tung y khi x=0). Trong th?ng k¨º ph??ng tr¨¬nh ???ng th?ng (b?c nh?t) n¨¤y ???c vi?t d??i d?ng: y= ?x + ? [1]
- 2. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:2 ??y l¨¤ ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh, trong ?¨® ? g?i l¨¤ ?? d?c (slope) v¨¤ ? l¨¤ ch?n (intercept), ?i?m c?t tr¨ºn tr?c tung khi x=0. Th?c ra ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh n¨¤y ch? c¨® tr¨ºn l? thuy?t, ngh?a l¨¤ c¨¢c tr? s? c?a xi (i=1,2,3,4,5,6) v¨¤ yi t??ng ?ng, li¨ºn h? v?i nhau 100% (ho?c h? s? t??ng quan R=1) Trong th?c t? hi?m khi c¨® s? li¨ºn h? 100% n¨¤y m¨¤ th??ng c¨® s? sai l?ch gi?a tr? s? quan s¨¢t yi v¨¤ tr? s? yi¡¯ ??c ?o¨¢n n?m tr¨ºn ???ng h?i qui. 17.1.1 M? h¨¬nh h?i qui tuy?n t¨ªnh V¨ª d? 2: Ta c¨® 1 m?u g?m 6 tr? em kh¨¢c c¨® c?n n?ng theo b?ng sau: Tu?i C?n n?ng (kg) 1 11 2 11 3 14 4 16 5 18 6 20 Khi v? ???ng th?ng h?i qui, ta th?y c¨¢c tr? s? quan s¨¢t y3, y4, y5, y6 n?m tr¨ºn ???ng th?ng, c¨°n y1 v¨¤ y2 kh?ng n?m tr¨ºn ???ng th?ng n¨¤y v¨¤ s? li¨ºn h? gi?a xi v¨¤ yi
- 3. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:3 kh?ng c¨°n l¨¤ 100% m¨¤ ch? c¨°n 97% v¨¬ c¨® s? sai l?ch t?i y1 v¨¤ y2. S? sai l?ch n¨¤y trong th?ng k¨º g?i l¨¤ ph?n d? (residual) ho?c errors. G?i y1, y2, y3, y4, y5, y6 l¨¤ tr? s? quan s¨¢t v¨¤ y¡¯1, y¡¯2, y¡¯3, y¡¯4, y¡¯5, y¡¯6 l¨¤ tr? s? ??c ?o¨¢n n?m tr¨ºn ???ng h?i qui, ?1, ?2, ?3, ?4, ?5, ?6 l¨¤ ph?n d?. Nh? v?y ?1= y1 ¨Cy¡¯1 ?2 = y2 ¨Cy¡¯2 ?3 = y3 ¨Cy¡¯3 ?4 = y4 ¨Cy¡¯4 ?5 = y5 ¨C y¡¯5 ?6 = y6 ¨Cy¡¯6 Khi ?¨® ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh ???c vi?t d??i d?ng t?ng qu¨¢t nh? sau: y¡¯= ¦Âxi + ?i+ ?i [2] Nh? v?y n?u ph?n d? ?i c¨¤ng nh? s? li¨ºn h? gi?a x,y c¨¤ng l?n v¨¤ ng??c l?i. Ph?n li¨ºn h? c¨°n ???i g?i l¨¤ ph?n h?i qui. M? h¨¬nh h?i qui tuy?n t¨ªch ???c m? t? nh? sau: D? li?u= H?i qui (Regression) + Ph?n d? (Residual) 17.1.2 ??c t¨ªnh h? s? t??ng quan ? v¨¤ ch?n ? Mu?n v? ???c ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh c?n ph?i ??c t¨ªnh ???c ?? d?c ? v¨¤ ch?n ? tr¨ºn tr?c tung. V¨ª d? 3: N?u ch¨²ng ta ch?n m?t m?u th?c t? g?m 30 em t? 1-6 tu?i v¨¤ k?t qu? c?n n?ng t??ng ?ng c?a 30 em ???c v? trong bi?u ?? sau:
- 4. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:4 L¨²c n¨¤y ta kh?ng th? n?i 30 ?i?m tr¨ºn bi?u ?? m¨¤ ph?i v? 1 ???ng th?ng ?i c¨¤ng g?n v?i t?t c? c¨¢c ?i?m c¨¤ng t?t. Nh? v?y 3 ???ng th?ng ? bi?u ?? ta ch?n ???ng th?ng n¨¤o?. Nguy¨ºn t?c ch?n ???ng th?ng n¨¤o ?i g?n c? 30 ?i?m, c¨® ngh?a l¨¤m sao ?? t?ng c¨¢c ph?n d? ??i nh? nh?t: ? ?i= ? (yi- ¦Âx ¨C ¦Á) v¨¤ t?ng b¨¬nh ph??ng c?a ph?n d?: ? (?i)2 = ? (yi- ¦Âx ¨C ¦Á)2 ??y l¨¤ ph??ng tr¨¬nh b?c 2 theo x. Trong to¨¢n h?c, mu?n t¨¬m tr? c?c ti?u c?a 1 ph??ng tr¨¬nh b?c 2, ng??i ta l?y ??o h¨¤m v¨¤ cho ??o h¨¤m tri?t ti¨ºu (b?ng 0) s? t¨¬m ???c tr? c?c ti?u c?a x. Gi?i ph??ng tr¨¬nh n¨¤y, ta s? t¨ªnh ???c 2 th?ng s? ? v¨¤ ? v¨¤ t? 2 th?ng s? n¨¤y ta s? v? ???c ???ng th?ng h?i qui. Ph??ng ph¨¢p n¨¤y trong to¨¢n h?c g?i l¨¤ ph??ng ph¨¢p b¨¬nh ph??ng nh? nh?t (least square method). Gi?i ph??ng tr¨¬nh tr¨ºn ta c¨®: ? = r Sy Sx (r l¨¤ h? s? t??ng quan; Sy l¨¤ ?? l?ch chu?n c?a y v¨¤ Sx l¨¤ ?? l?ch chu?n c?a x) r = 1 n-1 ? ( xi- x Sx ) ( yi- y Sy ) ?= y - ?x v¨¤ ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh c?a y theo x (b¨¬nh ph??ng nh? nh?t) l¨¤: y¡¯ = ¦Âxi + ? 17.2 Ph?n t¨ªch h?i qui tuy?n t¨ªnh trong SPSS Nh?p s? li?u tu?i v¨¤ c?n n?ng c?n ???c c?a 30 tr? 1-6 tu?i v¨¤o SPSS: C?t 1: tu?i; c?t 2: c?n n?ng
- 5. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:5 V¨¤o menu: >Analyze> Regression> Linear
- 6. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:6 B?ng 17.1 T¨®m t?t m? h¨¬nh H? s? t??ng quan R=0,918 v¨¤ R2 =0,843 B?ng 17. 2 Ph?n t¨ªch ANOVA v?i bi?n ph? thu?c l¨¤ c?n n?ng T?ng b¨¬nh ph??ng ph?n h?i qui (Regression)=336,14
- 7. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:7 T?ng b¨¬nh ph??ng ph?n d? (Residual)=62,8 Trung b¨¬nh b¨¬nh ph??ng h?i qui: 336,14/ 1 (b?c t? do)=336,14 Trung b¨¬nh b¨¬nh ph??ng ph?n d?: 62,8/ 28(b?c t? do=n-2)=2,24 F= 336,14 2,24 = 149,8 v¨¤ p<0,000 B?ng 17.3 H? s? t??ng quan ? v¨¤ ch?n ? K?t qu? b?ng 3 cho bi?t h? s? t??ng quan ? (?? d?c) = 1,96 v¨¤ ?i?m c?t t?i trung tung l¨¤ ?=7.773 Ph??ng tr¨¬nh ???ng th?ng h?i qui ???c vi?t: C?n n?ng= 7,77 + 1,96 x tu?i Nh? v?y khi em b¨¦ t?ng l¨ºn 1 tu?i th¨¬ c?n n?ng t?ng l¨ºn 1,96 kg V? ???ng th?ng h?i qui trong SPSS Linear Regression 1.00 2.00 3.00 4.00 5.00 6.00 tuoi 8.00 12.00 16.00 20.00 cannang ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? cannang = 7.77 + 1.96 * tuoi R-Square = 0.84
- 8. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:8 T? ph??ng tr¨¬nh n¨¤y ta c¨® th? ??c ?o¨¢n ???c c?n n?ng theo tu?i c?a tr?, tuy nhi¨ºn n?m trong m?t gi?i h?n n¨¤o ?¨® ch?ng h?n nh? t? 1-12 tu?i, v¨¬ sau tu?i n¨¤y c?n n?ng tr? s? t?ng v?t trong th?i k? d?y th¨¬ v¨¤ kh?ng c¨°n li¨ºn h? tuy?n t¨ªnh v?i tu?i n?a. V¨ª d? mu?n ??c ?o¨¢n c?n n?ng c?a tr? t? qu?n th? nghi¨ºn c?u n¨¤y: 7 tu?i ? C?n n?ng= 7,77 + 1,96 x7 = 21,49 kg 8 tu?i ? C?n n?ng= 7,77 + 1,96 x8 = 23,45 kg 17. 3 C¨¢c gi? ??nh trong ph?n t¨ªch h?i qui tuy?n t¨ªnh Ph?n t¨ªch h?i qui tuy?n t¨ªnh kh?ng ch? l¨¤ vi?c m? t? c¨¢c d? li?u quan s¨¢t ???c trong m?u (sample) nghi¨ºn c?u m¨¤ c?n ph?i suy r?ng cho m?i li¨ºn h? trong d?n s? (population). V¨¬ v?y, tr??c khi tr¨¬nh b¨¤y v¨¤ di?n d?ch m? h¨¬nh h?i qui tuy?n t¨ªnh c?n ph?i d¨° t¨¬m vi ph?m c¨¢c gi? ??nh. N?u c¨¢c gi? ??nh b? vi ph?m th¨¬ c¨¢c k?t qu? ??c l??ng kh?ng ?¨¢ng tin c?y ???c. C¨¢c gi? ??nh c?n thi?t trong h?i qui tuy?n t¨ªnh: 1. xi l¨¤ bi?n s? c? ??nh, kh?ng c¨® sai s¨®t ng?u nhi¨ºn trong ?o l??ng. 2. Ph?n d? (tr? s? quan s¨¢t tr? cho tr? s? ??c ?o¨¢n) ph?n ph?i theo lu?t ph?n ph?i chu?n 3. Ph?n d? c¨® tr? trung b¨¬nh b?ng 0 v¨¤ ph??ng sai kh?ng thay ??i cho m?i tr? xi 4. Kh?ng c¨® t??ng quan gi?a c¨¢c ph?n d? V¨ª d?: M?t nghi¨ºn c?u t¨¬m s? t??ng quan gi?a cholesterol m¨¢u v?i b? d¨¤y l?p n?i trung m?c (NTM) c?a ??ng m?ch c?nh ?o ???c tr¨ºn si¨ºu ?m v?i d? li?u ghi nh?n ? 100 BN nh? sau:
- 9. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:9 Bi?u ?? ph?n t¨¢n (scatter) l¨¤ m?t ph??ng ti?n t?t ?? ?¨¢nh gi¨¢ m?c ?? ???ng th?ng ph¨´ h?p v?i d? li?u quan s¨¢t. V¨¤o menu: Analyze> Curve Estimation
- 10. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:10 V¨¤o m¨¤n h¨¬nh Curve Estimation Nh?p chuy?n BEDAYNTM (B? d¨¤y n?i trung m?c) v¨¤o ? Dependent (s) v¨¤ CHOLESTEROL v¨¤o ? Variable. ?¨¢nh d?u nh¨¢y v¨¤o c¨¢c ? Include constant in equation, ? Plot models v¨¤ ? Linear (n?u mu?n ??c l??ng s? li¨ºn h? gi?a 2 bi?n theo d?ng ph??ng tr¨¬nh b?c 2 th¨¬ ?¨¢nh th¨ºm d?u nh¨¢y v¨¤o ? Quadratic). Nh?n OK, ta c¨® bi?u ?? sau:
- 11. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:11 ??y l¨¤ ph??ng tr¨¬nh h?i qui tuy?n t¨ªnh v?i y= 0,748 + 0,062x Gi? ??nh x ( cholesterol m¨¢u) l¨¤ m?t bi?n c? ??nh, kh?ng c¨® sai s¨®t trong ?o l??ng. Gi? ??nh n¨¤y kh?ng c¨® v?n ?? n?u b?nh nh?n ???c ?o ? m?t ph¨°ng th¨ª nghi?m chu?n. C¨¢c gi? ??nh c¨°n l?i th?c hi?n trong SPSS nh? sau: V¨¤o menu: Analyze> Regression> Linear...
- 12. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:12 V¨¤o m¨¤n h¨¬nh Linear, Nh?p chuy?n BEDAYNTM qua ? Dependent v¨¤ CHOLESTEROL qua ? Independent(s) Nh?n n¨²t Plots, m? h?p tho?i Plots: Nh?p chuy?n ph?n d? *ZRESID v¨¤o ? X (tr?c ho¨¤nh) v¨¤ gi¨¢ tr? d? ?o¨¢n v¨¤o ? Y (tr?c tung) ?? xem ph?n d? c¨® ph?n b? ng?u nhi¨ºn v¨¤ ph??ng sai c¨® c? ??nh cho m?i tr? c?a xi. Nh?n d?u nh¨¢y v¨¤o ? Histogram v¨¤ ? Normal probability plot ?? xem ph?n d? c¨® ph?n ph?i chu?n.
- 13. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:13 Nh?n Continue, sau ?¨® nh?n OK cho k?t qu? sau: Nh? v?y ph?n d? c¨® trung b¨¬nh (mean)=0 v¨¤ ?? l?ch chu?n (SD)=0,394 Bi?u ?? ph?n b? ph?n d? c¨® d?ng h¨¬nh chu?ng ??u 2 b¨ºn, tr? trung b¨¬nh g?n b?ng zero v¨¤ SD g?n b?ng 1. Nh? v?y gi? ??nh ph?n d? c¨® ph?n ph?i chu?n kh?ng b? vi ph?m. Ho?c xem bi?u ?? P-P plot so s¨¢nh gi?a ph?n ph?i t¨ªch l?y c?a ph?n d? quan s¨¢t (Observed Cum Prob) tr¨ºn tr?c ho¨¤nh v¨¤ ph?n ph?i t¨ªch l?y k?
- 14. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:14 v?ng (Expected Cum Prob) tr¨ºn tr?c tung. N?u c¨¢c ?i?m ??u n?m g?n ???ng ch¨¦o th¨¬ ph?n ph?i ph?n d? ???c coi nh? g?n chu?n. Cu?i c¨´ng ?? xem gi? ??nh c¨¢c ph??ng sai kh?ng ??i v?i m?i gi¨¢ tr? c?a x (cholesterol m¨¢u) ho?c g?i l¨¤ homoscedasticity. N?u c¨¢c tr? ph?n d? ph?n t¨¢n ng?u nhi¨ºn quanh gi¨¢ tr? zero (???ng ngang) th¨¬ coi nh? ph??ng sai kh?ng thay ??i, v¨¤ gi? ??nh v? homoscedasticity kh?ng b? vi ph?m.
- 15. TS Nguyen Ngoc Rang; Email: rangbvag@yahoo.com; Website: bvag.com.vn; Trang:15 N?u ph??ng sai thay ??i (l?n d?n ho?c nh? d?n theo gi¨¢ tr? c?a x) th¨¬ g?i l¨¤ Heteroscedascity (gi? ??nh v? ph??ng sai c? ??nh b? vi ph?m). V¨ª d? nh? h¨¬nh d??i ??y: T¨®m l?i, v?i v¨ª d? tr¨ºn c¨¢c gi? ??nh c?a ph?n t¨ªch h?i qui tuy?n t¨ªnh ??u th?a m?n v¨¤ ta c¨® th? k?t lu?n l¨¤ b? d¨¤y n?i trung m?c ??ng m?ch c?nh c¨® li¨ºn h? tuy?n t¨ªnh v?i n?ng ?? cholesterol m¨¢u theo ph??ng tr¨¬nh : Y (B? d¨¤y n?i trung m?c)= 0,062 X cholesterol + 0,748. Nh? v?y c? n?ng ?? cholesterol t?ng l¨ºn 1 mmol/L th¨¬ b? d¨¤y n?i trung m?c ??ng m?ch c?nh t?ng l¨ºn 0,062mm. T¨¤i li?u tham kh?o: 1. McClave J T and Sincich T. 2000. Simple linear regression in Statistics, 8th edition, Prentice-Hall, USA, pp. 505-557. 2. Moore D. S. and McCabe G. P. 1999. Looking at Data-Relationships (Chapter 2), in Introduction to the Practice of Statistics, W.H. Freeman and Company, New York, pp. 102-145.